Problem: Find $\dfrac{d}{dx}[-12\log_5(x)]$. Choose 1 answer: Choose 1 answer: (Choice A) A $-\dfrac{12}{\ln(x)5}$ (Choice B) B $-\dfrac{12}{5x}$ (Choice C) C $-\dfrac{12}{\ln(5)x}$ (Choice D) D $-\dfrac{12}{x}$
Solution: The expression to differentiate includes a logarithmic term. Remember that the derivative of the general logarithmic term $\log_a(x)$ (where $a$ is any positive constant and $a\neq 1$ ) is $\dfrac{1}{\ln(a)\cdot x}$. Put another way, $\dfrac{d}{dx}[\log_a(x)]=\dfrac{1}{\ln(a)\cdot x}$. [Is there an easy way to memorize that?] We can use this to find the derivative as shown below. $\begin{aligned} &\phantom{=}\dfrac{d}{dx}[-12\log_5(x)] \\\\ &=-12\dfrac{d}{dx}[\log_5(x)] \\\\ &=-12\cdot\dfrac{1}{\ln(5)x} \\\\ &=-\dfrac{12}{\ln(5)x} \end{aligned}$ In conclusion, $\dfrac{d}{dx}[-12\log_5(x)]=-\dfrac{12}{\ln(5)x}$.